ANALYSISOF EXPERIMENTAL TERMENERGIES to their effect on the oxidation of Cr" by HzOzand air. Gaps in our knowledge of the exact distribution of the various oxidation states of chlorine in the radiolytic decomposition of Clod- do not affect the general arguments given above, since what was said applies to all possible products of decomposition of Clod-. I n concluding, instead of giving a summary of the findings reported in the paper, we give a more or less equivalent rBsum6 of the procedure to be followed in the application of the Cr''-Cr''' system for the study of unknown intermediates. The acid used is HC104 at low concentrations (-0.2 M should be all right). Higher concentrations of HC104, as well as other acids such as HzS04,lead to formation of complex ions which
2335 may introduce unnecessary complications unless a competition study is desired. The complexes formed are separated and are identified; then their individual G values are determined a t various initial conditions. Finally, we assess the importance of the thermal or radiation-induced hydrolysis and ligand reactions or the formation of the Cr"' complexes in order to establish their quantitative correspondence to the intermediates.
Acknowledgment. This work was performed under the auspices of the Greek Atomic Energy Commission. The authors wish to thank Dr. A. 0.Allen and Professor A. C. Wahl for helpful suggestions.
Analysis of Experimental Term Energies' by Norman Padnos Department of Chemistry, North Carolina College at Durham, Durham, North Carolina
(Received August 25, 1967)
If the energy of a state be known for several isoelectronic monatomic species, the average potential energy of interaction of one electron, or of several electrons, with the nucleus and with the remaining electrons can be estimated. For one electron, these are, respectively: En1= Z(dT/dZ) and Eel = 22' - Enl. T is the energy of the state considered, taking as zero the energy of the species with the electron completely removed, and 2 is the atomic number. The above expressions are derived using the Hellmann-Feynman theorem and the virial theorem and neglecting magnetic interactions (spin-orbit, etc.).
Introduction A criterion which is commonly used in evaluating an approximate atomic wave function is agreement of the energy calculated from the wave function with the spectroscopically observed energy. The calculated energy can be analyzed into contributions from the kinetic energy, the electron-nucleus potential energy E, = x ( Z e 2 / r t ) , and the electron-electron potential i
energy E, = C(e2/r,,). The virial theorem enables one to analyze the observed energy into contributions from kinetic energy and from potential energy. The present work develops and illustrates a method of analyzing the potential energy thus derived from observation into electron-nucleus and electron-electron contributions. Thus three energy quantities, kinetic energy, E,, and E,, derived from any approximate wave function can be compared to experimental data. We start with the Hamiltonian
where the indices i and j refer to electrons. The Hellmann-Feynman theorem2 states that
"=($)=qz2) bZ
Thus, the nuclear contribution to the energy
bE E, = 2 ( - Z e 2 / r , ) = ZbZ
(11)
This is the total energy of attraction of the nucleus for all the electrons in the atom if E represents the energy required to remove all the electrons. If only the term energy, T, be known, we still have T = E Eionoland
-
This is approximately (Ze2/r) for the electron re(1) This work was supported in part by a Dupont Summer Grant for Teachers in Chemistry. (2) (a) R. P.Feynman, Phy8. Rev., 56,340 (1939); (b) H. Heflmann, "Einfuhrung in die Quantenchemie," Franz Deuticke, Leipzig, 1m7. Volume 78, Number 7
July 1068
NORMAN PADNOS
2336 Table I : N e I Series
-T ,
d4/-T/dZa
kK
Ne I N a I1 Mg I11
173,932 381,528 646 364 967 783 1 345,100
A1 1V Si V
417.05 617.68 803.97 983.76 1159.78
211.69 191.73 182.16 177.88
1766 2604 3515 4550
1.24 1.41 1.58 1.74
1.111 1.222 1.333 1.444
...
0.658 0.490 0.396 0.332
0.97 0.66 0.51
Estimated using a five-point Lagrange formula.6
moved to form the I n this way, we can estimate, from the observed energy of a state in an isoelectronic series, the potential energy of an electron in the field of the nucleus. The virial theorem4 states that, for the Hamiltonian I , V = 2E, where V is the potential energy, and E is the total energy. Since the potential energy is the sum of the electron-nucleus and electron-electron contributions, we have for the latter
E,’
=
E,
- Eeo(ion) = 2T - Z(dT/dZ)
(IV)
Using the same sort of approximation as before, we can interpret this as the energy of interaction of the electron removed, with the other electrons. I n expressions I11 and IV, the derivative, dT/dZ, is not strictly measurable since we can only make measurements at integral values of 2. An estimate can be made, however, from a formula for term values such as6
(T/R)’/*= (Z
- a)/n
or by numerical differentiation of a set of d a h 6 Probably the best estimate can be obtained by a combination of these procedures, finding 1/?1and differentiating this more nearly linear function numerically. This is the procedure followed here.
Calculations In order to illustrate the use of the formulas given and to see if they give reasonable results, we have estimated En*, Be1,(En’/Ee’I, and a mean radius ( l / ~ ) - ’ for some species. We would expect that lEn’/Ee’l be of the order of the ratios of the atomic number t o the number of electrons less one, Z / ( n - l), e.g., for Li I1 3/1 = 3.0, for He I 2/1, for Pd I1 46/44 = 1.045. (l/r)-l should be less than the radius of the species. One-ElectronIons. This case is trivial, since E, = 0, but serves as a check on the theory developed here. The observed energies are
T
=
Z2R/n2
The calculated En is ZdT/dZ = Z(2ZR/n2) = 2T in accord with the virial theorem, and the calculated 2T = 0, also the correct value.
E, is 2T
-
The Journal of Physical Chcmhtry
Ions Isoelectronic with Ne I : the Ground States. The ground state of these ions is 2pe(lS). They ionize t o give ~ P ~ ( ~ Table P ) . I gives the ionization energies’& and the square root of the ionization energy for the first five members of this series and calculated quantities for the first four members. The calculated radii are smaller than the (crystal) ionic radii and vary in the same sense from Na I1 to A1 IV. The jEn /E e ratios are consistently larger than the charge ratios and vary in the same manner. Po? II; the 4 ~ l * ( ~ F5s) ( 4 F ) State. The ionization energies from this state for Rh I, Pd 11, and Ag I11 are’C98 -58.71, -129.53, and -215.01 kK. En is -3660 kK and E, is 3410 kK. The ratio lEnl/Eellis l.OS, compared with Z / ( n - 1) = 1.045. He I and Li 11: Ground State, 2@S), 4 s ( W ) , 4d(3D) Xtates. The ionization energies7a for He I, I,i 11, and
’I
Table 11: He I and Li I1
He I Is2
Li I1 lsa Be I11 1st
He I 2s Li I1 2s Be I11 2s He I 4s Li I1 4s He I 4d Li I1 4d
198.305 610.097 1241.2 38.45 134.03 248.7 8.013 30.097 6.866 27.467
14.08 24.70 35.23 6.20 11.58 16.87 2.831 5.486 2.620 5.241
10.66 10.57
2.95 4.52
2.00 3.00
5.42 5.33
2.34 3.62
2.00 3.00
2.655 2.655 2.621 2.621
2.14 3.23 2.00 3.00
2.00 3.00 2.00 3.00
(3) This approximation should be most accurate when the electron is in a highly excited state to begin with, as its removal in this case would not be expected to greatly change the other ~ 1 ’ s . (4) H. Eyring, J. Walter, and G. E. Kimball, “Quantum Chemistry,” John Wiley and Sons, Inc., New York, N. Y., 1944, pp 355-358; W. Kauzmann, “Quantum Chemistry,” Academic Press, New York, N. Y., 1957, pp 229-233. (5) H. E. White, “Introduction to Atomic Spectra,” McGraw-Hill Book Co., Inc., New York, N. Y., 1934, Chapter 17. (6) 2. Kopal, “Numerical Analysis,” 2nd ed, John Wiley and Sons Inc., New York, N. Y., 1961, pp 87-92,554-556. (7) C. E. Moore, “Atomic Energy Levels,” National Bureau of Standards Circular 467, U. S. Government Printing Office, Washington, D. C.: (a) Vol. I, 1949; (b) Vol. 11, 1952; (0) Vol. 111, 1958. (8) The values given are averaged over the four possible J values for the quartet states.
THEPHOTOCHEMISTRY OF GASEOUS ACETONE Be I11 were used to evaluate the derivatives for the first two states, but data on He I and Li I1 only were used for the other states. Table I1 shows data and
2337 calculated quantities. From Table 11, it is evident that, as the electron becomes more excited, the ratio, lEnl/Eellapproaches Z/(n - 1).
The Photochemistry of Gaseous Acetone by Henry Shaw and Sidney Toby School of Chemistry, Rutgers University, New Brunswick, New Jersey
08905
(Received August 28, 1967)
The photochemistry of acetone was reinvestigated in the temperature range 121-298' and the pressure range 0.1-220 torr. The quantum yield of carbon monoxide production was independent of pressure over the large pressure range studied. Data were obtained giving the variation of quantum yields of methane, ethane, methyl ethyl ketone, 2,5-hexanedione, and a number of minor products as a function of pressure and temperature, and a mechanism is proposed which accounts for the major products. The behavior of methane formation a t low pressures suggests that intramolecular formation of methane is important under some conditions. The quantum yield for acetone disappearance was unity at low pressures but increased rapidly a t pressures of acetone greater than 10 torr. After taking energy-transfer considerations into account, excellent 2 agreement was obtained with published values of l~2/k,'/~for the reactions CHa A + CHd CHaCOCHz CzH~*. By correcting literature values of k,, the high-pressure limit was found to be k,, = and 2CH3 (2.0 f 0.15)10QT'/21. mol-1 sec-l. This yields k$ = (3.3 f 1.5)10*exp[(-9440 f 350)/RT] 1. mol-1 sec-1.
+
Introduction The photolysis of gaseous acetone (A) has been extensively studied because i t is one of the principle sources of quantitative data on the kinetics of methyl radical reactions. The mechanism was postulated by Dorfman and Noyes' and may be written
+ h~ -+ CH3 + CHIC0 CHIC0 (+ A) A CH3 + CO (+ A) CHa + A -%- CH4 + CHaCOCHz CHa + CH3COCHz -% C2H6COCHa A
2CHa
3C2Ha* b
+
+
CZHO* A -% CzHB A Reactions a, b, and c were shown to be important at low pressures by Dodd and Steacie,Z and the asterisk indicates vibrationally excited e thane. More recently, Darwent, Allard, Hartman, and Langea determined that reaction 2 was not sufficient to account for all of the methane produced in the photolysis of acetone above 200". They speculated on the additional abstraction reaction CHa
+ CH3COCHZ
CH4
+ CHzCOCHz
+
but they did not obtain any direct evidence for this reaction. Henderson and Steacie4 also observed more methane than would be predicted from reaction 2 only. They attributed this methane to a reaction of methyl radicals with excited acetone. They presented strong evidence to show that the source of this additional methane was not due to methyl radicals abstracting from ethane. Ausloos and Steacie5 clearly demonstrated that a t 27" additional methane was produced from the reaction CH3
+ CH3CO A CH, + CHzCO
They observed ketene in their product gases and found a product dependency on the square root of the incident intensity. O'Neal and Bensone demonstrated that the acetyl radical is sufficiently long lived bo react with hydrogen iodide a t temperatures above 200". The photolysis of gaseous acetone a t low pressures is (1) L. M. Dorfman and W. A. Noyes, Jr., J . Chem. Phys., 16, 557 (1948). (2) R. E. Dodd and E. W. R. Steacie, Proc. Roy. SOC. (London), A223, 283 (1954). (3) B. deB. Darwent, M. J. Allard, M. F. Hartman, and L. J. Lange, J . Phys. Chem., 64, 1847 (1960). (4) J. F.Henderson and E. W. R. Steacie, Can. J . Chem., 38, 2161 (1960). (5) P.Ausloos and E. W. R. Steacie, ibid., 33, 47 (1955). (6) E. O'Neal and S. W. Benson, J . Chem. Phys., 36, 2196 (1962). Volume 7.2, Number 7 July 1068
